Let z₁, z₂ be the roots of the equation z² + az + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, the value of |a| is :

Solution

Maths sir · 4 steps

Use the equilateral triangle condition with one vertex at 0, then turn it into sum and product of roots.

1. Equilateral condition. Points z₁, z₂, z₃ form an equilateral triangle exactly when z₁² + z₂² + z₃² = z₁ z₂ + z₂ z₃ + z₃ z₁. Here the third vertex is the origin, so put z₃ = 0.

   z₁² + z₂² = z₁ z₂ ⇒ (z₁ + z₂)² = 3 z₁ z₂

2. Use Vieta. From z² + az + 12 = 0 you get z₁ + z₂ = -a and z₁ z₂ = 12. Substituting gives a² = 3 × 12 = 36.
3. Take modulus. a may be complex, so take the modulus of both sides: |a|² = |a²| = 36.
4. So. |a| = √36, which is 6.

The common slip: 36. Stopping at a² = 36 and reporting that value gives 36. The question asks for |a|, which is its square root, 6.

The idea to keep. When one vertex of an equilateral triangle is the origin, the condition becomes z₁² + z₂² = z₁ z₂, which is (z₁+z₂)² = 3z₁z₂. That form plugs straight into sum and product of roots.

Answer from the NTA final answer key. Solution by PrepWiser.